arXiv research

The paper classifies degenerate almost complex surfaces in a nearly Kähler space.

New geometries derived from symplectic Monge-Ampère structures.

Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle

There exist non-degenerate 3-form $dω_I$, $ω_I(X,Y)=g(IX,Y)$, for each leftinvariant almost Hermitian structure $(g,I)$, where $g$ is Killing-Cartan metric on the $M=S^3\times S^3=SU(2)\times SU(2)$. Known \cite{H1}, that arbitrary non-degenerate 3-form on the 6-dimensional manifold, with some additional properties def…

Let $(M,I)$ be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (so-called Nijenhuis tensor) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold is called nearly Kaehler if it adm…

The paper proves smoothness of almost-minimizers' boundaries near the free boundary.

An almost complex manifolds $(M^4,J)$ of real dimension 4 with non-degenerate torsion bundle admit a double absolute parallelism and it is provided the classification of homogeneous $(M^4,J)$ having an associated non-solvable Lie algebra. We extend such a classification to the analysis of the manifolds having an associ…

The paper discovers new ways Riemann surfaces can degenerate.

Defines log Floer cohomology for symplectic surfaces with a degenerate part.

Defines pre-Kähler structures and their properties.

Maps converge to simpler structures under certain tension conditions.

Constructs a family to handle unstable fibers on complex surfaces.

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual…

We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homo…

Study of complex surfaces in a specific pseudo-Riemannian space.

In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. The study of these hypersurfaces is motivated by the fact that for indefinite almost Hermitian manifolds this class of lightlike hypersurfaces does not exist. We also establish that radical transversal…

Generically an almost complex structure has no symmetries at all, but there exist symmetric structures. In this paper we describe how to guarantee that the pseudogroup of local symmetries is small (finite-dimensional). It will be indicated that a large symmetry pseudogroup (infinite-dimensional) is a signature of some …

In this paper we study submanifolds of an almost complex manifold with Norden metric which are non-degenerate with respect to the one Norden metric and lightlike with respect to the other Norden metric on the manifold. Relations between the induced geometric objects of some of these submanifolds are given. Examples of …

The paper explores conditions for the existence of orthogonal almost complex structures on manifolds.

We correct an error in the second part of Theorem 3 of our original paper (arXiv:1512.07161).

Lectures on symplectic aspects of surface degenerations at KIAS.

Study shows limits of volume-constrained sets are finite unions of Wulff shapes.

Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.

In this paper almost complex surfaces of the nearly Kähler $S^3\times S^3$ are studied in a systematic way. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product structure on the nearly Kähler $S^3\times S^3$. We also find a correspondence betwe…

Researchers prove smoothings for surfaces with triple points.

We study almost complex surfaces in the nearly Kähler $S^3\times S^3$. We show that there is a local correspondence between almost complex surfaces and solutions of the H-surface equation introduced by Wente. We find a global holomorphic differential on every almost complex surface, and show that when this differentia…

Study degenerations of Kähler-Einstein metrics on surfaces.

Introduces generalized moment maps for almost Hermitian settings.

Let (M,ω) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,ω) that ensure that the r…

Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.

We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition …

Classifies degenerations of complex projective plane with rational singularities.

We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations for the almost contact metric structure. To make the theory meaningful, we cons…

We define a non-degenerated Monge-Ampere structure on a 6-manifold associated with a Monge-Ampere equation as a couple (Ω,ω), such that Ωis a symplectic form and ωis a 3-differential form which satisfies ω\wedgeΩ=0 and which is non-degenerated in the sense of Hitchin. We associate with such a couple an almost (pseudo) …

The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.

We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of $(\mathbb{R}^4,J)$, for some almost complex structure $J$ if and only if it is an elliptic curve. Furthermore we show that any (almost) complex $2n$-torus can be holomorphically embedded in $(\mathbb{R}^{4n},J)$ for a suitable almo…

Study Hermitian metrics with Bismut connection satisfying Bianchi identity and SKT condition.

The study examines harmonic forms on almost Hermitian manifolds and complex surfaces.

Study of quasi-Kähler metrics on complex manifolds linked to c-projective metrizability.

Study pseudo-Kähler structures on almost abelian solvmanifolds.

In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equatio…

In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provi…

In this paper we will prove that for a compact, symplectic manifold $(M, ω)$ and for $ω$-compatible almost-complex structure J any properly perturbed J-holomorphic curve has a non-negative symplectic area. This non-negative property provides us with a new obstruction to the bubbling off phenomenon and thus allows us to…

New criterion for almost-complex 4-manifolds using polyhedral decompositions.

New algebraic structures for Hermitian geometry cohomologies.

Study Bergman kernel metrics on degenerating hyperelliptic surfaces.

Study of pseudoconvex 3-manifolds in complex surfaces.